Integrand size = 23, antiderivative size = 331 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=-\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 \sqrt {1+(c+d x)^2}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b^3 d (a+b \text {arcsinh}(c+d x))}+\frac {e^2 \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{24 b^4 d}-\frac {9 e^2 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^4 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 b^4 d} \]
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Time = 0.45 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5859, 12, 5779, 5818, 5778, 3384, 3379, 3382, 5773, 5819} \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{24 b^4 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 b^4 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 b^4 d}-\frac {3 e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{2 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {e^2 \sqrt {(c+d x)^2+1}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b d (a+b \text {arcsinh}(c+d x))^3} \]
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5773
Rule 5778
Rule 5779
Rule 5818
Rule 5819
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^2 x^2}{(a+b \text {arcsinh}(x))^4} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int \frac {x^2}{(a+b \text {arcsinh}(x))^4} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}+\frac {\left (2 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^3} \, dx,x,c+d x\right )}{3 b d}+\frac {e^2 \text {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^3} \, dx,x,c+d x\right )}{b d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arcsinh}(c+d x))^2}+\frac {e^2 \text {Subst}\left (\int \frac {1}{(a+b \text {arcsinh}(x))^2} \, dx,x,c+d x\right )}{3 b^2 d}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {x^2}{(a+b \text {arcsinh}(x))^2} \, dx,x,c+d x\right )}{2 b^2 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 \sqrt {1+(c+d x)^2}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b^3 d (a+b \text {arcsinh}(c+d x))}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \left (-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{2 b^4 d}+\frac {e^2 \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))} \, dx,x,c+d x\right )}{3 b^3 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 \sqrt {1+(c+d x)^2}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {e^2 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{3 b^4 d}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b^4 d}-\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b^4 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 \sqrt {1+(c+d x)^2}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b^3 d (a+b \text {arcsinh}(c+d x))}+\frac {\left (e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{3 b^4 d}-\frac {\left (3 e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b^4 d}+\frac {\left (9 e^2 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b^4 d}-\frac {\left (e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{3 b^4 d}+\frac {\left (3 e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b^4 d}-\frac {\left (9 e^2 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{8 b^4 d} \\ & = -\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d (a+b \text {arcsinh}(c+d x))^3}-\frac {e^2 (c+d x)}{3 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 (c+d x)^3}{2 b^2 d (a+b \text {arcsinh}(c+d x))^2}-\frac {e^2 \sqrt {1+(c+d x)^2}}{3 b^3 d (a+b \text {arcsinh}(c+d x))}-\frac {3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b^3 d (a+b \text {arcsinh}(c+d x))}+\frac {e^2 \text {Chi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{24 b^4 d}-\frac {9 e^2 \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^4 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{8 b^4 d} \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.78 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\frac {e^2 \left (-\frac {8 b^3 (c+d x)^2 \sqrt {1+(c+d x)^2}}{(a+b \text {arcsinh}(c+d x))^3}+\frac {4 b^2 \left (-2 (c+d x)-3 (c+d x)^3\right )}{(a+b \text {arcsinh}(c+d x))^2}-\frac {4 b \sqrt {1+(c+d x)^2} \left (2+9 (c+d x)^2\right )}{a+b \text {arcsinh}(c+d x)}-80 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )+80 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )+27 \left (3 \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )\right )\right )}{24 b^4 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(708\) vs. \(2(307)=614\).
Time = 0.47 (sec) , antiderivative size = 709, normalized size of antiderivative = 2.14
method | result | size |
derivativedivides | \(\frac {\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{2} \left (9 b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+18 a b \,\operatorname {arcsinh}\left (d x +c \right )-3 b^{2} \operatorname {arcsinh}\left (d x +c \right )+9 a^{2}-3 a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arcsinh}\left (d x +c \right )+a^{3}\right )}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{4}}-\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )-b^{2} \operatorname {arcsinh}\left (d x +c \right )+a^{2}-a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arcsinh}\left (d x +c \right )+a^{3}\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{48 b^{4}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{24 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{3}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{24 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d}\) | \(709\) |
default | \(\frac {\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{2} \left (9 b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+18 a b \,\operatorname {arcsinh}\left (d x +c \right )-3 b^{2} \operatorname {arcsinh}\left (d x +c \right )+9 a^{2}-3 a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arcsinh}\left (d x +c \right )+a^{3}\right )}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{4}}-\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{2} \left (b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+2 a b \,\operatorname {arcsinh}\left (d x +c \right )-b^{2} \operatorname {arcsinh}\left (d x +c \right )+a^{2}-a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}+3 a \,b^{2} \operatorname {arcsinh}\left (d x +c \right )^{2}+3 a^{2} b \,\operatorname {arcsinh}\left (d x +c \right )+a^{3}\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (d x +c \right )+\frac {a}{b}\right )}{48 b^{4}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{24 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{3}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (d x +c \right )-\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{24 b \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d}\) | \(709\) |
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\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]
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\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=e^{2} \left (\int \frac {c^{2}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx\right ) \]
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Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\text {Timed out} \]
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\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^4} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4} \,d x \]
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